The Number System: Rational and Irrational Numbers, Roots, and Estimation
The Number System strand on the Grade 8 exam tests how fluently you move between fractions, terminating and repeating decimals, roots, and irrational numbers. Expect items on classifying numbers, converting repeating decimals to fractions, estimating square and cube roots without a calculator, placing irrational values on a number line, and comparing quantities such as √50 and 7.1.
Rational vs. irrational
A rational number can be written as a/b where a and b are integers and b ≠ 0. Its decimal expansion either terminates (3/8 = 0.375) or eventually repeats (2/11 = 0.181818…). An irrational number cannot be written that way; its decimal never terminates and never settles into a repeating block. √2, √3, π, and 0.101001000… are irrational.
Every positive number has two square roots, but the radical sign √ denotes the principal (nonnegative) root, so √49 = 7, not ±7. Cube roots of negatives are negative: ∛(−64) = −4.
Estimating roots
Bound the root between consecutive perfect squares or cubes. Since 49 < 50 < 64, √50 lies between 7 and 8. Refine by squaring candidates: 7.1² = 50.41 is too large, and 7.05² = 49.7025 is too small, so √50 ≈ 7.07.
To compare two positive irrational values, square both: √50 versus 7.05 becomes 50 versus 49.7025, so √50 > 7.05.
Converting repeating decimals
Let x = 0.272727… Then 100x = 27.272727…, so 100x − x = 27, giving 99x = 27 and x = 27/99 = 3/11. The number of repeating digits determines the power of ten you multiply by.
Worked example
Order √50, 7.05, and 22/3 from least to greatest. 22/3 = 7.333… and √50 ≈ 7.071. Now compare 7.05 with √50 by squaring: 7.05² = 49.7025 < 50, so 7.05 < √50. The order is 7.05 < √50 < 22/3.
Traps
- Computing √50 as 25, or treating the radical as 'divide by 2.'
- Assuming √(a + b) = √a + √b. It does not: √(9 + 16) = 5, not 7.
- Calling every radical irrational; √81 = 9 is rational.
- Using 22/7 as though it equals π. It is only an approximation.
- Rounding intermediate estimates before the final comparison, which can flip the order.